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Consider the Ideal Gas Law

PV = ​kT,

where k>0

is a constant. Solve this equation for V in terms of P and T.

​a) Determine the rate of change of the volume with respect to the pressure at constant temperature. Interpret the result.

​b) Determine the rate of change of the volume with respect to the temperature at constant pressure. Interpret the result.

​c) Assuming k =1,

draw several level curves of the volume function and interpret the results.

1 Answer

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Answer and explanation:

Given : Consider the Ideal Gas Law,
PV=kT where k>0 is a constant.

To find : Solve this equation for V in terms of P and T.

Solution :


PV=kT

Divide each side by P,


V=(kT)/(P) ....(1)

a) Determine the rate of change of the volume with respect to the pressure at constant temperature. Interpret the result.

Differentiate equation (1) w.r.t P,


(dV)/(dP)=kT(d)/(dP)((1)/(P))


(dV)/(dP)=kT(-(1)/(P^2))


(dV)/(dP)=-(kT)/(P^2)

​b) Determine the rate of change of the volume with respect to the temperature at constant pressure. Interpret the result.

Differentiate equation (1) w.r.t T,


(dV)/(dT)=(k)/(P)(d)/(dT)(T)


(dV)/(dP)=(k)/(P)(1)


(dV)/(dP)=(k)/(P)

c) Assuming k =1, draw several level curves of the volume function and interpret the results.

When k=1,
PV=T


(dV)/(dP)=-(T)/(P^2) <0


(dV)/(dP)=(1)/(P) >0

Refer the attached figure below.

Consider the Ideal Gas Law PV = ​kT, where k>0 is a constant. Solve this equation-example-1
User Mauro Stepanoski
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